1. Chapter 0 Functions

2. § 0.1
Functions and Their Graphs

3. Rational & Irrational Numbers
Definition
Example
Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers
Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern

4. The Number Line The Number Line
A geometric representation of the real numbers is shown below.

5. Open & Closed Intervals
Definition
Example
Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves
Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves
[-1, 4]

6. Functions
EXAMPLE
If , find f (a - 2).

7. Domain Definition Example
Domain of a Function: The set of acceptable values for the variable x.
The domain of the function is

8. Graphs of Functions Definition Example
Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xy-plane.

9. The Vertical Line Test Definition Example
Vertical Line Test: A curve in the xy-plane is the graph of a function if and only if each vertical line cuts or touches the curve at no more than one point.
Although the red line intersects the graph no more than once (not at all in this case), there does exist a line (the yellow line) that intersects the graph more than once. Therefore, this is not the graph of a function.

10. Graphing Using a Graphing Calculator
Graphing Calculators
Graphing Using a Graphing Calculator
Step
Display
1) Enter the expression for the function.
2) Enter the specifications for the viewing window.
3) Display the graph.

11. Graphs of Equations EXAMPLE
Is the point (3, 12) on the graph of the function ?

12. § 0.2
Some Important Functions

13. Linear Equations y = mx + b x = a Equation Example
(This is a linear function)
x = a
(This is not the graph of a function)

14. Linear Equations
CONTINUED
Equation
Example
y = b

15. Piece-Wise Functions EXAMPLE
Sketch the graph of the following function

16. Quadratic Functions Quadratic Function: A function of the form
Definition
Example
Quadratic Function: A function of the form
where a, b, and c are constants and a 0.

17. Polynomial Functions Polynomial Function: A function of the form
Definition
Example
Polynomial Function: A function of the form
where n is a nonnegative integer and a0, a1, ...an are given numbers.

18. Rational Functions
Definition
Example
Rational Function: A function expressed as the quotient of two polynomials.

19. Power Functions Power Function: A function of the form Definition
Example
Power Function: A function of the form

20. Absolute Value Function
Definition
Example
Absolute Value Function: The function defined for all numbers x by
such that |x| is understood to be x if x is positive and –x if x is negative

21. § 0.4
Zeros of Functions – The Quadratic Formula and Factoring

22. Zeros of Functions
Definition
Example
Zero of a Function: For a function f (x), all values of x such that f (x) = 0.

23. Quadratic Formula
Definition
Example
Quadratic Formula: A formula for solving any quadratic equation of the form
The solution is:
There is no solution if
These are the solutions/zeros of the quadratic function

24. Graphs of Intersecting Functions
EXAMPLE
Find the points of intersection of the pair of curves.

25. Factoring
EXAMPLE
Factor the following quadratic polynomial.

26. Factoring
EXAMPLE
Solve the equation for x.

27. § 0.5
Exponents and Power Functions

28. Exponents
Definition
Example
bn = b*b*b…*b

29. Exponents
Definition
Example

30. Exponents
Definition
Example

31. Exponents
Definition
Example

32. Exponents
Definition
Example

33. Applications of Exponents
EXAMPLE
Use the laws of exponents to simplify the algebraic expression.

34. Compound Interest - Annual
Definition
Example
Compound Interest Formula:
A = the compound amount (how much money you end up with)
P = the principal amount invested
i = the compound interest rate per interest period
n = the number of compounding periods
If $700 is invested, compounded annually at 8% for 8 years, this will grow to:
Therefore, the compound amount would be $1,

35. Compound Interest - General

36. Compound Interest - General
EXAMPLE
(Quarterly Compound) Assume that a $500 investment earns interest compounded quarterly. Express the value of the investment after one year as a polynomial in the annual rate of interest r.